Tech Manual Excerpt
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FILMTEC Membranes
System Design: System Performance Projection
Design Equations
and Parameters
The performance of a specified RO system is defined by its feed pressure (or permeate
flow, if the feed pressure is specified) and its salt passage. In its simplest terms, the
permeate flow Q through an RO membrane is directly proportional to the wetted surface
area S multiplied by the net driving pressure (∆P – ∆π). The proportionality constant is the
membrane permeability coefficient or A-value. The familiar water permeation equation has
the form:
Q = ( A)(S )(∆P − ∆π )
Eq. 8
The salt passage is by diffusion, hence the salt flux NA is proportional to the salt
concentration difference between both sides of the membrane. The proportionality constant
is the salt diffusion coefficient or B-value.
Eq. 9
N A = B(C fc − C p )
where:
Cfc = feed-concentrate average concentration
Cp = permeate concentration
There are basically two ways to calculate the performance of a specified design: “Elementto-Element” and “Entire System”.
Element-to-Element
This is the most rigorous calculation method. It is too tedious for hand calculation, but it is
suitable for computer calculations. All the operating conditions of the first element must be
known, including the feed pressure. Then the flow, pressure, etc., of the concentrate, which
is the feed to the second element, can be calculated. After calculating the results for all the
elements, the original feed pressure may be too high or low, so the trial and error process
starts with a new pressure.
With the help of the FILMTECTM Reverse Osmosis System Analysis (ROSA) computer
program, accurate results can be obtained very quickly, so this program can be used to
modify and optimize the design of an RO or an NF system. Accordingly, the entire system
calculation method will not be described here. It is also not intended to outline the process
of the element to element computer calculation. However, the governing equations and
parameters are given in Table 3.9.
In order to enable the determination of values for the terms A, ∆P, and ∆π in Eq. 8, the
water permeation equation is expanded to Eq. 10. The permeate concentration can be
derived from Eq. 9 after conversion into Eq. 19. The design equations are listed in Table
3.9, the symbol definitions in Table 3.11.
Page 1 of 4
* Trademark of The Dow Chemical Company
Form No. 609-02057-604
Design Equations
and Parameters
(cont.)
The subscript i in the equations of Table 3.9 indicates that they apply to the i th element in a
sequence of n elements in a series flow configuration. To accurately determine system
performance, Eq. 10 is successively solved for each of the n elements starting with an inlet
set of conditions. The solutions depend on mass balances around each element for salt
(Eq. 14) and water (Eq. 19), as well as correlations for individual element parameters such
as concentrate-side flow resistance, ∆Pfc (Eq. 27c); temperature correction factor for water
permeability, TCF (Eq. 16); polarization factor, pfi (Eq. 17), and the membrane permeability
coefficient for water, Ai (πi) (Eq. 28) which in the case of the FILMTEC FT30 membrane
depends on the average concentrate concentration or, alternatively, osmotic pressure.
These solutions usually involve a suitable average for the feed and permeate side hydraulic
and osmotic pressures. For low recovery values typical of single element operation, an
accurate solution can be obtained using a simple arithmetic average of the inlet and outlet
conditions. Even so, since the outlet conditions are not known, iterative trial and error
solutions are involved.
Table 3.9 Design equations for projecting RO system performance: individual
element performance
Item
Permeate flow
Equation
Average concentrate-side
osmotic pressure
⎛C
πi = π fi ⎜⎜ fc i
⎝ Cf i
Average permeate-side osmotic
pressure
Ratio: arithmetic average
concentrate-side to feed
concentration for Element i
π pi = π fi (1 − R i )
Ratio: concentrate to feed
concentration for Element i
Cc i 1 − Yi (1 − Ri )
=
(1 − Yi )
Cf i
Feed water osmotic pressure
π f = 1.12(273 + T )
∆Pfc i
⎛
⎞
− Ppi − π + π pi ⎟⎟
Qi = Ai πi SE (TCF )(FF )⎜⎜ Pfi −
2
⎝
⎠
Temperature correction factor for
RO and NF membrane
Concentration polarization factor
for FILMTEC 8-inch elements
System recovery
11
⎞
⎟(pfi )
⎟
⎠
Cfc i 1 ⎛ Cc i
= ⎜⎜ 1 +
Cf i
2⎝
Cf i
Equation
Number
10
12
13
⎞
⎟
⎟
⎠
14
∑m
15
j
⎡
1
⎛ 1
TCF = EXP⎢2640⎜
−
⎝ 298 273 + T
⎣
⎞⎤
⎟⎥; T ≥ 25°C
⎠⎦
⎡
1
⎛ 1
TCF = EXP⎢3020⎜
−
298
273
+T
⎝
⎣
⎞⎤
⎟⎥; T ≤ 25°C
⎠⎦
pfi = EXP[0.7Yi ]
16a,b
17
Y = 1 − [(1 − Y1 )(1 − Y2 )...(1 − Yn )] = 1 −
n
∏ (1 − Yi )
18
i =1
Permeate concentration
Page 2 of 4
( )
C p j = B Cfc j (pfi )(TCF )
* Trademark of The Dow Chemical Company
SE
Qi
19
Form No. 609-02057-604
Design Equations
and Parameters
(cont.)
Entire System
Average values are used to calculate feed pressure and permeate quality if the feed quality,
temperature, permeate flow rate and number of elements are known. If the feed pressure is
specified instead of the number of elements, the number of elements can be calculated with
a few iterations. The design equations for 8-inch BW30 FILMTEC elements are listed in
Table 3.10, the symbol definitions in Table 3.11.
Table 3.10 Design equations for projecting RO system performance: system average
performace
Equation
Number
Item
Total permeate flow
Equation
Ratio: average concentrateside to feed concentration
for system
Limiting system recovery
Cfc
− R ln(1 − Y / YL )
=
+ 1− R
Cf
Y − (1 − YL )ln(1 − Y / YL )
Q = NE SE A π (TCF)(FF)Pf −
Cfc
Cf
( )( )
)
20
21
22
Pf − ∆P fc − Pp
ln(1 − Y )
Y
=−
YL ,R =1
Yi = 1 − (1 − Y )
1/ n
[
pf = EXP 0.7Yi
Average concentrate-side
osmotic pressure for system
⎛C
π = π i ⎜⎜ fc
⎝ Cf
Individual FILMTEC 8-inch
element, or single-stage
concentrate-side pressure
drop
FILMTEC membrane
permeability as a function of
average concentrate-side
osmotic pressure
)
π f pf R
Average polarization factor
Average concentrate-side
system pressure drop for
FILMTEC 8-inch elements;
2 stages
(
(
YL = 1 −
Approximate log-mean
concentrate-side to feed
concentration ratio for
system
Average element recovery
⎤
⎡C
∆P fc
Pp − π f ⎢ fc pf − 1 − R ⎥
2
C
⎦
⎣ f
]
24
25
26
⎞
⎟ pf
⎟
⎠
∆P fc = 0.04q fc
23
2
27a,b,c
⎡ 0.1(Q / 1440)⎤⎛ 1
⎞
∆Pfc = ⎢
+ 1 − Y ⎟⎟
⎥⎜⎜
YN
N
V2
⎣
⎦⎝ VR
⎠
∆Pfc = 0.01nq fc
1.7
A (π ) = 0.125 ; π ≤ 25
28a,b,c
⎛ π − 25 ⎞
A (π ) = 0.125 − 0.011⎜
⎟ ; 25 ≤ π ≤ 200
⎝ 35 ⎠
A (π ) = 0.070 − 0.0001(π − 200); 200 ≤ π ≤ 400
Permeate concentration
Page 3 of 4
⎛N S ⎞
C p = BCfc pf (TCF )⎜ E E ⎟
⎝ Q ⎠
* Trademark of The Dow Chemical Company
29
Form No. 609-02057-604
Design Equations
and Parameters
Table 3.11 Symbol definitions
(cont.)
Q
i
∑
permeate flow of Element i (gpd)
summation of all ionic species
j
Ai π i
membrane permeability at 25° for Element i, a function of
the average concentrate-side osmotic pressure (gfd/psi)
SE
membrane surface area per element (ft2)
Y
n
∏
system recovery (expressed as a fraction) = permeate
flow/feed flow
multiplication of n terms in a series
i =1
TCF
temperature correction factor for membrane
permeability
n
number of elements in series
FF
membrane fouling factor
Q
system permeate flow (gpd)
Pf i
feed pressure of Element i (psi)
NE
number of elements in system
∆Pfc i
concentrate-side pressure drop for Element i (psi)
Pp i
permeate pressure of Element i (psi)
πi
average concentrate-side osmotic pressure (psi)
πf i
feed osmotic pressure of Element i
R
average fractional salt rejection for system
πpi
permeate-side osmotic pressure of Element i (psi)
π
average concentrate-side osmotic pressure for system (psi)
pfi
concentration polarization factor for Element i
∆Pfc
average concentrate-side system pressure drop (psi)
Ri
salt rejection fraction for Element i
YL
limiting (maximum) system recovery (expressed as a
fraction)
=
Qi
Aπ
Cfc
feed conc. - perm. conc.
feed conc.
average element permeate flow (gpd) = Q/NE
average membrane permeability at 25°C: a function of the
average concentrate-side osmotic pressure (gfd/psi)
average concentrate-side concentration for system (ppm)
Cfc i
average concentrate-side concentration for Element i (ppm) Y
i
average element recovery (expressed as a fraction)
Cf i
feed concentration for Element i (ppm)
pf
average concentration polarization factor
Cc i
concentrate concentration for Element i (ppm)
q fc
arithmetic average concentrate-side flow rate (gpm)
(=1/2(feed flow + concentrate flow)
Yi
recovery fraction for Element i
NV
number of six-element pressure vessels in system (≈ NE/6)
=
permeate flow
feed flow
πf
treated feed water osmotic pressure (psi)
NV 1
number of pressure vessels in first stage of 2-stage system
(≈ 1/3 NV)
T
feed water temperature (°C)
NV 2
number of pressure vessels in second stage of 2-stage
system (≈ NV/3)
mj
molal concentration of jth ion species
NVR
stage ratio (=NVI/NV2)
FILMTEC Membranes
Notice: The use of this product in and of itself does not necessarily guarantee the removal of cysts and pathogens from water.
Effective cyst and pathogen reduction is dependent on the complete system design and on the operation and maintenance of
the system.
Notice: No freedom from any patent owned by Seller or others is to be inferred. Because use conditions and applicable laws
may differ from one location to another and may change with time, Customer is responsible for determining whether products
and the information in this document are appropriate for Customer’s use and for ensuring that Customer’s workplace and
disposal practices are in compliance with applicable laws and other governmental enactments. Seller assumes no obligation or
liability for the information in this document. NO WARRANTIES ARE GIVEN; ALL IMPLIED WARRANTIES OF
MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE ARE EXPRESSLY EXCLUDED.
Page 4 of 4
*Trademark of The Dow Chemical Company
Form No. 609-02057-604